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Mixture models

What are mixture models?

Mixture models are a combination of two or more distributions added together to create a distribution that has a shape with more flexibility than a single distribution. Each of the mixture’s components must be multiplied by a proportion, and the sum of all the proportions is equal to 1. The mixture is generally written in terms of the PDF, but since the CDF is the integral (cumulative sum) of the PDF, we can equivalently write the Mixture model in terms of the PDF or CDF. For a mixture model with 2 distributions, the equations are shown below:

\({PDF}_{mixture} = p\times{PDF}_1 + (1-p)\times{PDF}_2\)

\({CDF}_{mixture} = p\times{CDF}_1 + (1-p)\times{CDF}_2\)

\({SF}_{mixture} = 1-{CDF}_{mixture}\)

\({HF}_{mixture} = \frac{{PDF}_{mixture}}{{SF}_{mixture}}\)

\({CHF}_{mixture} = -ln({SF}_{mixture})\)

Mixture models are useful when there is more than one failure mode that is generating the failure data. This can be recognised by the shape of the PDF and CDF being outside of what any single distribution can accurately model. On a probability plot, a mixture of failure modes can be identified by bends or S-shapes in the data that you might otherwise expect to be linear. An example of this is shown in the image below. You should not use a mixture model just because it can fit almost anything really well, but you should use a mixture model if you suspect that there are multiple failure modes contributing to the failure data you are observing. To judge whether a mixture model is justified, look at the goodness of fit criterion (AICc or BIC) which penalises the score based on the number of parameters in the model. The closer the goodness of fit criterion is to zero, the better the fit. Using AD or log-likelihood for this check is not appropriate as these goodness of fit criterions do not penalise the score based on the number of parameters in the model and are therefore prone to overfitting.

See also competing risk models for another method of combining distributions using the product of the SF rather than the sum of the CDF.

_images/mixture_required1.png

Creating a mixture model

Within reliability.Distributions is the Mixture_Model. This function accepts an array or list of standard distribution objects created using the reliability.Distributions module (available distributions are Exponential, Weibull, Gumbel, Normal, Lognormal, Loglogistic, Gamma, Beta). There is no limit to the number of components you can add to the mixture, but is is generally preferable to use as few as are required to fit the data appropriately (typically 2 or 3). In addition to the distributions, you can specify the proportions contributed by each distribution in the mixture. These proportions must sum to 1. If not specified the proportions will be set as equal for each component.

As this process is additive for the survival function, and may accept many distributions of different types, the mathematical formulation quickly gets complex. For this reason, the algorithm combines the models numerically rather than empirically so there are no simple formulas for many of the descriptive statistics (mean, median, etc.). Also, the accuracy of the model is dependent on xvals. If the xvals array is small (<100 values) then the answer will be “blocky” and inaccurate. The variable xvals is only accepted for PDF, CDF, SF, HF, and CHF. The other methods (like random samples) use the default xvals for maximum accuracy. The default number of values generated when xvals is not given is 1000. Consider this carefully when specifying xvals in order to avoid inaccuracies in the results.

The API is similar to the other probability distributions (Weibull, Normal, etc.) and has the following inputs and methods:

Inputs:

  • distributions - a list or array of probability distributions used to construct the model.
  • proportions - how much of each distribution to add to the mixture. The sum of proportions must always be 1.

Methods:

  • name - ‘Mixture’
  • name2 - ‘Mixture using 3 distributions’
  • mean
  • median
  • mode
  • variance
  • standard_deviation
  • skewness
  • kurtosis
  • excess_kurtosis
  • b5 - The time where 5% have failed. Same as quantile(0.05)
  • b95 - The time where 95% have failed. Same as quantile(0.95)
  • plot() - plots all functions (PDF,CDF,SF,HF,CHF)
  • PDF() - plots the probability density function
  • CDF() - plots the cumulative distribution function
  • SF() - plots the survival function (also known as reliability function)
  • HF() - plots the hazard function
  • CHF() - plots the cumulative hazard function
  • quantile() - Calculates the quantile (time until a fraction has failed) for a given fraction failing. Also known as b life where b5 is the time at which 5% have failed.
  • inverse_SF() - the inverse of the Survival Function. This is useful when producing QQ plots.
  • mean_residual_life() - Average residual lifetime of an item given that the item has survived up to a given time. Effectively the mean of the remaining amount (right side) of a distribution at a given time.
  • stats() - prints all the descriptive statistics. Same as the statistics shown using .plot() but printed to console.
  • random_samples() - draws random samples from the distribution to which it is applied.

Example 1

The following example shows how the Mixture_Model object can be created, visualised and used.

from reliability.Distributions import Lognormal_Distribution, Gamma_Distribution, Weibull_Distribution, Mixture_Model
import matplotlib.pyplot as plt

# create the mixture model
d1 = Lognormal_Distribution(mu=2, sigma=0.8)
d2 = Weibull_Distribution(alpha=50, beta=5, gamma=100)
d3 = Gamma_Distribution(alpha=5, beta=3, gamma=30)
mixture_model = Mixture_Model(distributions=[d1, d2, d3], proportions=[0.3, 0.4, 0.3])

# plot the 5 functions using the plot() function
mixture_model.plot()

# plot the PDF and CDF
plot_components = True # this plots the component distributions. Default is False
plt.figure(figsize=(9, 5))
plt.subplot(121)
mixture_model.PDF(plot_components=plot_components, color='red', linestyle='--')
plt.subplot(122)
mixture_model.CDF(plot_components=plot_components, color='red', linestyle='--')
plt.subplots_adjust(left=0.1, right=0.95)
plt.show()

# extract the mean of the distribution
print('The mean of the distribution is:', mixture_model.mean)

'''
The mean of the distribution is: 74.91607709895453
'''
_images/Weibull_Mixture_distV1.png _images/Weibull_Mixture_dist_propsV1.png

Fitting a mixture model

Within reliability.Fitters is Fit_Weibull_Mixture. This function will fit a Weibull Mixture Model consisting of 2 x Weibull_2P distributions (this does not fit the gamma parameter). Just as with all of the other distributions in reliability.Fitters, right censoring is supported, though care should be taken to ensure that there still appears to be two groups when plotting only the failure data. A second group cannot be made from a mostly or totally censored set of samples.

Whilst some failure modes may not be fitted as well by a Weibull distribution as they may be by another distribution, it is unlikely that a mixture of data from two distributions (particularly if they are overlapping) will be fitted noticeably better by other types of mixtures than would be achieved by a Weibull mixture. For this reason, other types of mixtures are not implemented.

Inputs:

  • failures - an array or list of the failure data. There must be at least 4 failures, but it is highly recommended to use another model if you have less than 20 failures.
  • right_censored - an array or list of right censored failure data. Optional input.
  • print_results - True/False. This will print results to console. Default is True
  • CI - confidence interval for estimating confidence limits on parameters. Must be between 0 and 1. Default is 0.95 for 95% CI.
  • optimizer - ‘L-BFGS-B’, ‘TNC’, or ‘powell’. These are all bound constrained methods. If the bounded method fails, nelder-mead will be used. If nelder-mead fails then the initial guess will be returned with a warning. For more information on optimizers see scipy.
  • show_probability_plot - True/False. This will show the probability plot with the fitted mixture CDF. Default is True.

Outputs:

  • alpha_1 - the fitted Weibull_2P alpha parameter for the first (left) group
  • beta_1 - the fitted Weibull_2P beta parameter for the first (left) group
  • alpha_2 - the fitted Weibull_2P alpha parameter for the second (right) group
  • beta_2 - the fitted Weibull_2P beta parameter for the second (right) group
  • proportion_1 - the fitted proportion of the first (left) group
  • proportion_2 - the fitted proportion of the second (right) group. Same as 1-proportion_1
  • alpha_1_SE - the standard error on the parameter
  • beta_1_SE - the standard error on the parameter
  • alpha_2_SE - the standard error on the parameter
  • beta_2_SE - the standard error on the parameter
  • proportion_1_SE - the standard error on the parameter
  • alpha_1_upper - the upper confidence interval estimate of the parameter
  • alpha_1_lower - the lower confidence interval estimate of the parameter
  • beta_1_upper - the upper confidence interval estimate of the parameter
  • beta_1_lower - the lower confidence interval estimate of the parameter
  • alpha_2_upper - the upper confidence interval estimate of the parameter
  • alpha_2_lower - the lower confidence interval estimate of the parameter
  • beta_2_upper - the upper confidence interval estimate of the parameter
  • beta_2_lower - the lower confidence interval estimate of the parameter
  • proportion_1_upper - the upper confidence interval estimate of the parameter
  • proportion_1_lower - the lower confidence interval estimate of the parameter
  • loglik - Log Likelihood (as used in Minitab and Reliasoft)
  • loglik2 - LogLikelihood*-2 (as used in JMP Pro)
  • AICc - Akaike Information Criterion
  • BIC - Bayesian Information Criterion
  • AD - Anderson-Darling goodness of fit statistic
  • results - a dataframe of the results (point estimate, standard error, Lower CI and Upper CI for each parameter)
  • goodness_of_fit - a dataframe of the goodness of fit values (Log-likelihood, AICc, BIC, AD).

Example 2

In this example, we will create some data using two Weibull distributions and then combine the data using np.hstack. We will then fit the Weibull mixture model to the combined data and will print the results and show the plot. As the input data is made up of 40% from the first group, we expect the proportion to be around 0.4.

from reliability.Fitters import Fit_Weibull_Mixture
from reliability.Distributions import Weibull_Distribution
from reliability.Other_functions import histogram
import numpy as np
import matplotlib.pyplot as plt

# create some failures from two distributions
group_1 = Weibull_Distribution(alpha=10, beta=3).random_samples(40, seed=2)
group_2 = Weibull_Distribution(alpha=40, beta=4).random_samples(60, seed=2)
all_data = np.hstack([group_1, group_2])  # combine the data
results = Fit_Weibull_Mixture(failures=all_data) #fit the mixture model

# this section is to visualise the histogram with PDF and CDF
# it is not part of the default output from the Fitter
plt.figure(figsize=(9, 5))
plt.subplot(121)
histogram(all_data)
results.distribution.PDF()
plt.subplot(122)
histogram(all_data, cumulative=True)
results.distribution.CDF()

plt.show()

'''
Results from Fit_Weibull_Mixture (95% CI):
Analysis method: MLE
Failures / Right censored: 100/0 (0% right censored)

    Parameter  Point Estimate  Standard Error  Lower CI  Upper CI
      Alpha 1         8.65512        0.393835   7.91663   9.46249
       Beta 1         3.91196        0.509775   3.03021   5.05029
      Alpha 2         38.1103         1.41076   35.4432   40.9781
       Beta 2         3.82192        0.421384   3.07917   4.74385
 Proportion 1        0.388491       0.0502663  0.295595  0.490263

Goodness of fit    Value
 Log-likelihood -375.991
           AICc  762.619
            BIC  775.007
             AD 0.418645
'''
_images/Weibull_Mixture_V6.png _images/Weibull_Mixture_histV2.png

Example 3

In this example, we will compare how well the Weibull Mixture performs vs a single Weibull_2P. Firstly, we generate some data from two Weibull distributions, combine the data, and right censor it above our chosen threshold. Next, we will fit the Mixture and Weibull_2P distributions. Then we will visualise the histogram and PDF of the fitted mixture model and Weibull_2P distributions. The goodness of fit measure is used to check whether the mixture model is really a much better fit than a single Weibull_2P distribution (which it is due to the lower BIC).

from reliability.Fitters import Fit_Weibull_Mixture, Fit_Weibull_2P
from reliability.Distributions import Weibull_Distribution
from reliability.Other_functions import histogram, make_right_censored_data
import numpy as np
import matplotlib.pyplot as plt

# create some failures and right censored data
group_1 = Weibull_Distribution(alpha=10, beta=2).random_samples(700, seed=2)
group_2 = Weibull_Distribution(alpha=30, beta=3).random_samples(300, seed=2)
all_data = np.hstack([group_1, group_2])
data = make_right_censored_data(all_data, threshold=30)

# fit the Weibull Mixture and Weibull_2P
mixture = Fit_Weibull_Mixture(failures=data.failures, right_censored=data.right_censored, show_probability_plot=False, print_results=False)
single = Fit_Weibull_2P(failures=data.failures, right_censored=data.right_censored, show_probability_plot=False, print_results=False)
print('Weibull_Mixture BIC:', mixture.BIC, '\nWeibull_2P BIC:', single.BIC) # print the goodness of fit measure

# plot the Mixture and Weibull_2P
histogram(all_data, white_above=30)
mixture.distribution.PDF(label='Weibull Mixture')
single.distribution.PDF(label='Weibull_2P')
plt.title('Comparison of Weibull_2P with Weibull Mixture')
plt.legend()
plt.show()

'''
Weibull_Mixture BIC: 6431.578404093574
Weibull_2P BIC: 6511.51175959736
'''
_images/Weibull_mixture_vs_Weibull_2P_V5.png